2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis B
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Masaharu Tanabe
- Class Format
- Lecture (Livestream)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon
- Class
- -
- Course Code
- MTH.C402
- Number of credits
- 100
- Course offered
- 2022
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
Lectures are a sequel to ''Advanced topics of Analysis A'' in the previous quarter.
A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations.
We will study the most important theorems concerning closed Riemann surfaces,
in this course, Abel’s theorem and the Jacobi inversion theorem.
Using these and the Riemann-Roch theorem a topic in the previous quarter, we will study the Jacobian varieties and holomorphic maps of closed Riemann surfaces.
Course description and aims
At the end of this course, students are expected to understand the main classical results of the theory of closed Riemann surfaces, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion.
Keywords
Riemann surfaces, Abel’s theorem, the Jacobi inversion theorem, Jacobian varieties
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Applications of the Riemann-Roch Theorem I (Weierstrass points) | Details will be provided during each class session. |
| Class 2 | Applications of the Riemann-Roch Theorem II (automorphisms of closed Riemann surfaces) | |
| Class 3 | Abel’s theorem | |
| Class 4 | The Jacobi inversion theorem | |
| Class 5 | The Jacobian varieties I | |
| Class 6 | The Jacobian varieties II | |
| Class 7 | Holomorphic maps of closed Riemann surfaces |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None in particular
Reference books, course materials, etc.
H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag
Evaluation methods and criteria
Assignments. Details will be announced during the session.
Related courses
- ZUA.C301 ： Complex Analysis I
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
- MTH.C401 ： Advanced topics in Analysis A
Prerequisites
Students are expected to have completed Advanced topics in Analysis A (MTH.C401).
Other
None in particular